Quantitative Aptitude Formulae
1. Number system
2. H.C.F & L.C.M of Numbers
3. Ages
4. Averages
5. Percentage
6. Profit & Loss
7. Ratio & Proportion
8. Mixtures and alligations
9. Partnership
10. Chain Rule
11. Time & Work
12. Pipes & Cisterns
13. Time And Distance
14. Trains
15. Boats & Streams
16. Simple Interest
17. Compound Interest
18. Races and Games
Number System
1. A number is divisible by 2, if its unit’s place digit is 0, 2, 4, or 8
2. A number is divisible by 3, if the sum of its digits is divisible by 3
3. A number is divisible by 4, if the number formed by its last two digits is divisible by 4
4. A number is divisible by 6, if the number is divisible by 2 and 3.
5. A number is divisible by 7, if the difference of a number obtained by removing the last digit of a number and double the last digit of a number. Ex: 91 check it like 9 – 2(1) = 7
6. A number is divisible by 8, if the number formed by its last three digits is divisible by 8
7. A number is divisible by 9, if the sum of its digits is divisible by 9
8. A number is divisible by 11, if, starting from the RHS, (Sum of its digits at the odd place) – (Sum of its digits at even place) is equal to 0 or multiple of 11.
9. (a + b)2 = a2+ 2ab + b2
10. (a - b)2 = a2- 2ab + b2
11. (a + b)2 - (a - b)2= 4ab
12. (a + b)2 + (a - b)2 = 2(a2 + b2)
13. (a2 – b2)= (a + b)(a - b)
14. (a3 + b3)= (a + b)(a2 - ab + b2)
15. (a3 – b3)= (a - b)(a2 + ab + b2)
16. Results on Division:
Dividend = Quotient × Divisor + Remainder
17. An Arithmetic Progression (A. P.) with first term ‘a’ and Common Difference ‘d’ is given
by:
[a], [(a + d)], [(a + 2d)], … … …, [a + (n - 1)d]
nth term, Tn = a + (n - 1)d
Sum of first ‘n’ terms, Sn = n/2 (First Term + Last Term) or (n/2)(2a + (n-1)d)
18. A Geometric Progression (G. P.) with first term ‘a’ and Common Ratio ‘r’ is given by:
a, ar, ar2, ar3, … … …, arn-1
nth term, Tn = arn-1
Sum of first ‘n’ terms Sn = [a(1 - rn)] / [1 - r]
19. (1 + 2 + 3 + … … … + n) = [n(n + 1)] / 2
20. (12 + 22+ 32 + … … … + n2) = [n(n + 1)(2n + 1)] / 6
21. (13 + 23+ 33 + … … … + n3) = [n2(n + 1)2] / 4
H.C.F & L.C.M of Numbers
Product of two numbers = Their H. C. F. × Their L. C. M.
Problems on Ages
We need to remember three things:
Past/ Ago/ Back/ Before : x -
Present : x
Future/Hence/Later/ After : x +
Working methodology: In these problems, two persons initial ages will be given. And before or after several years, their ratio of the ages will be given. Multiply the ratio of their initial age by x or some variable and take them as their initial age. Now if final ratio has been given, equate this ratio with that ratio and find x. Or proceed according to the problem.
Averages
1. Average or mean: The Mean (Average) of a group of numbers is the sum of the numbers divided by the number of numbers:
Average or Mean = Sum of the observations / Number of observations
2. If the average of ‘m ’ quantities is ‘x ‘ and the average age of ‘ n ‘ other quantities is ‘y ‘ then the average of all of them put together is = mx+nym+nmx+nym+n
3. If the average age of ‘m ‘ quantities is ‘x ‘ and the average age of ‘n ’ quantities out of them (m quantities) is ‘ y ‘ then the average of the rest of the quantities is = .mx−nym−nmx−nym−n
4. If the average of ‘ n ‘ numbers is ‘ x ’ and if ‘ k ‘ is added to or subtracted from each given number the average of ‘ n ‘ numbers becomes (x+k) or (x-k) respectively. In the other words average value will be increased or decreased by ‘ k ‘.
5. If the average of ‘ n ‘ numbers is ‘ x ‘ and if each given number is multiplied to or divided by ‘ k ‘ then the average of n numbers becomes kx or xkxk respectively.
6. If a person travels a distance at a speed of x km/hr and the same distance at a speed of y km/hr then the average speed during the whole journey is given by 2xyx+y2xyx+y km/hr.
If a person covers A km at x km/hr and B km at y km/hr and C km at z km/hr, then the average speed in covering the whole distance is A+B+CAx+By+CzA+B+CAx+By+Czkm/hr.
7. The average of n (where n is an odd number) consecutive numbers is always the middle number E.g. 1, 3, 5, 7, 9.
The Average = Middle number = 5
8. The average ‘ n ‘ (where n is even number) consectuive numbers is the average of the two middle numbers.
E.g. Average of (2, 4, 6, 8, 10, 12) = 6+826+82 = 7
Percentages
1. To express x% as a fraction, we have x% = x / 100
2. To express a / b as a percent, we have a / b = (a / b × 100) %
3. If ‘A’ is R% more than ‘B’, then ‘B’ is less than ‘A’ by
Or
If the price of a commodity increases by R%, then the reduction in consumption, not to increase the expenditure is {100R / [100 + R] } %
4. If ‘A’ is R% less than ‘B’, then ‘B’ is more than ‘A’ by
Or
If the price of a commodity decreases by R%, then the increase in consumption, not to increase the expenditure is {100R / [100 - R] } %
5. If a number is successively decreased by x% and y%, then the single equivalent percentage decrease in that number will be ((-x) +(- y) + ((-x)(-y)/100))
6. If a number is successively increased by x% and y%, then the single equivalent percentage increase in that number will be (x + y + (xy/100))
7.If the population of a town is ‘P’ in a year, then its population after ‘N’ years is
P (1 + R/100)N
P / [(1 + R/100)N]
Profit & Loss
1. Profit(P) = SP – CP, Loss(L) = CP – SP
2. Profit or Gain % = (P/CP) * 100, Loss % = (L/CP) * 100
3. SP = [(100 + Gain%) × CP] / 100 or [(100 - Loss%) × Cost Price] / 100
4. CP = (SP * 100) / (100 + Gain%) or (SP * 100) / (100 - Loss%)
5. If the value of a machine is ‘P’ in a year, then its value ‘N’ years ago at a depreciation of ‘R’ p.c.p.a is P / [(1 - R/100)N]
6. If the value of a machine is ‘P’ in a year, then its value after ‘N’ years at a depreciation of ‘R’ p.c.p.a is P (1 - R/100)N
Ratio & Proportion
Ø The equality of two ratios is called a proportion. If a : b = c : d, we write a : b :: c : d and we say that a, b, c, d are in proportion.
Ø In a proportion, the first and fourth terms are known as extremes, while the second and third are known as means.
1. Product of extreme terms = Product of mean terms
2. Mean proportion between A and B is √AB
3. The compounded ratio of the ratios (a : b), (c : d), (e : f) is (ace : bdf)
4. a2 : b2 is a duplicate ratio of a : b
5. √a : √b is a sub-duplicate ration of a : b
6. a3 : b3 is a triplicate ratio of a : b
7. a1/3 : b1/3 is a sub-triplicate ratio of a : b
8. If a / b = c / d, then, (a + b) / b = (c + d) / d, which is called the componendo.
9. If a / b = c / d, then, (a - b) / b = (c - d) / d, which is called the dividendo.
10. If a / b = c / d, then, (a + b) / (a - b) = (c + d) / (c - d), which is called the componendo & dividendo.
Alligation or Mixture
Alligation: It is the rule that enables us to find the ratio in which two or more ingredients at the given price must be mixed to produce a mixture at a given price.
Mean Price: The cost price of a quantity of the mixture is called the mean price.
Rule of Alligation: If two ingredients are mixed, then:
We represent the above formula as under:
(i) When investments of all the partners are for the same time, the gain or loss is distributed among the partners in the ratio of their investments.
Suppose A and B invest Rs. x and Rs. y respectively for a year in a business, then at the end of the year : (A’s share of profit) : (B’s share of profit) = x : y.
(ii) When investments are for different time periods, then equivalent capitals are calculated for a unit of time by taking (capitalnumber of units of time). Now, gain or loss is divided in the ratio of these capitals.
Suppose A invests Rs. x for ‘p’ months and B invests Rs. y for ‘q’ months, then (A’s share of profit) : (B’s share of profit) = xp : yq.
(III) When investments are altered in the given period we need to take the changes into consideration while calculating their profits.
Suppose A and B started their business with Rs.5000 and Rs.10,000 respectively. If after three months A invested another Rs.5000 then we have to consider A's capital for the remaining period is Rs.10,000
So A: B = (5000 X 3 + 10,000 X 9) : 10,000 X 12 = 1,05,000 : 1,20,000 = 7:8
3. Working and Sleeping Partners : A partner who manages the business is known as a working partner and the on who simply invests the money is a sleeping parnter.
Chain Rule
1. The work done is directly proportional to the number of men working at it.
2. The time (number of days) required to complete a job is inversely proportional to the number of hours per day allocated to the job.
3. Time taken to cover a distance is inversely proportional to the speed of the car.
The Relation between Number of Days(D), Number of persons(M). Amount of time in hrs or minutes or seconds(T) and amount of work(W) is (M1 * D1 * T1) / W1 = (M2 * D2 * T2 ) / W2
Time & Work
1. If A can do a piece of work in n days, then A’s 1 day’s work = 1/n.
2. If A’s 1 day’s work = 1/n, then A can finish the work in n days.
3. If A is thrice as good a workman as B, then:
Ratio of work done by A and B = 3 : 1,
Ratio of times taken by A & B to finish a work = 1 : 3
Pipes & Cisterns
1. If a pipe can fill a tank in ‘x’ hours and another pipe can empty the full tank in ‘y’ hours (where y > x), then on opening both the pipes, the net part of the tank filled in 1 hour is (1/x – 1/y)
Time And Distance
1. Distance = Speed × Time
2. 1 km/hr = 518518 m/s
3. y metres/sec = (y × 18/5) km/hr
4. If the ratios of speed is a : b : c, then the ratios of time taken is : 1a:1b:1c1a:1b:1c
Relation between variables T, D, S:
To distance, Speed and Time both are directly proportional and
To speed, Time is inversely proportional.
if speed is doubled, distance covered in a given time also gets doubled and ,
if speed is doubled, time taken to cover a distance will be half.
Average speed is defined as =.Total distance travelled / Total time taken
Problems on Trains
Relative Speed:
1. If two objects are moving in opposite directions towards each other at speeds u and v, then relative speed = Speed of first + Speed of second = u + v.
2. If the two objects move in the same direction with speeds u and v, then
relative speed = difference of their speeds = u – v.
Model 1. 1 Pole and I Train:
Length of The Train (m) = Speed of the Train (m/s) × Time taken to cross the pole (s)
Model 2. 1 Train and 1 Platform:
Length of the Train + Length of the Platform (m) = Speed of the Train (m/s) × Time taken to cross the platform (s)
Model 3. 1 Train with speed speed v1 and 1 moving person with speed v2
Case 1: If both are moving in same direction
Length of The Train (m) = [Speed of the Train - Speed of the Man] (m/s) × Time taken to cross the man (s)
Case 2: If both are moving in opposite direction
Length of The Train (m) = [Speed of the Train + Speed of the Man] (m/s) × Time taken to cross the man (s)
Model 4. 2 Trains with speeds v1,v2
Case 1: If both are moving in same direction
[Length of The Train 1 + Length of the Train 2](m) = [Speed of the Train1 - Speed of the Train 2] (m/s) × Time taken to cross (s)
Case 2: If both are moving in opposite direction
[Length of The Train 1 + Length of the Train 2](m) = [Speed of the Train1 + Speed of the Train 2] (m/s) × Time taken to cross (s)
Boats and Streams
1. If the speed of a boat in still water is u km/hr and the speed of the stream is v hm/hr, then:
Speed downstream = (u + v) km/hr. Speed upstream = (u - v) km/hr.
2. If the speed downstream is a km/hr and the speed upstream is b km/hr, then:
Speed in still water = ½ (a + b) km/hr.
Rate of stream = ½ (a - b) km/hr.
Simple Interest
Let Principle = P, Rate = R% per annum and Time = T years. Then,
a. S.I. = ( P × R × T ) / 100
b. P = ( 100 × S.I. ) / ( R × T ),
c. R = ( 100 × S.I. ) / ( P × T ),
d. T = ( 100 × S.I. ) / ( P × R ).
Compound Interest
1. Let Principle = P, Rate = R% per annum and Time = T years. Then,
I. When interest is compounded Annually, Amount = P (1 + R/100)N
II. When interest is compounded Half-yearly: Amount = P (1 + R/2/100)2N
III. When interest is compounded Quarterly: Amount = P (1 + R/4/100)4N
2. When Rates are different for different years, say R1%, R2%, R3% for 1st, 2nd, and 3rd year respectively,
Then, Amount = P (1 + R1/100) (1 + R2/100) (1 + R3/100)
Races and Games
When all participants reach the finishing point at the same instant of time, the race is said to end in a "Dead Heat"
The various types of phrases used in problems on races and their interpretations are as follows:
1. A gives B y meters: This means, both A and B start at the starting point at the same instant of time, but while A reaches the finishing point, B is y meters behind. This indicates that A is the winner of the race.
2. A gives B t minutes: This means, bothe A and B start at the starting point at the same instant, but B takes t minutes more as compared to A to finish the race. Here also, A is the winner.
3. A can give B a start of y meters: A starts from the starting point and B starts y meters ahead, but still both A and B reach the finishing point at the same instant of time. So, the race ends in a dead heat.
4. A can give B a start of t minutes: A starts t minutes after B starts from the starting point, but still, both A and B reach the finishing point at the same instant of time. So, again the race ends in a dead heat.
5. A gives B y meters and t minutes: A and B start at the starting point at the same instant, but while A reaches the finishing point, B is behind by y meters,and, B takes t minutes compared to A to complete the race. So, B covers remaining y meters y meters in extra t minutes. This gives the speed of B as y/t